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Mirrors > Home > ILE Home > Th. List > infrenegsupex | Unicode version |
Description: The infimum of a set of reals is the negative of the supremum of the negatives of its elements. (Contributed by Jim Kingdon, 14-Jan-2022.) |
Ref | Expression |
---|---|
infrenegsupex.ex | |
infrenegsupex.ss |
Ref | Expression |
---|---|
infrenegsupex | inf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lttri3 7851 | . . . . . 6 | |
2 | 1 | adantl 275 | . . . . 5 |
3 | infrenegsupex.ex | . . . . 5 | |
4 | 2, 3 | infclti 6910 | . . . 4 inf |
5 | 4 | recnd 7801 | . . 3 inf |
6 | 5 | negnegd 8071 | . 2 inf inf |
7 | negeq 7962 | . . . . . . . . 9 | |
8 | 7 | cbvmptv 4024 | . . . . . . . 8 |
9 | 8 | mptpreima 5032 | . . . . . . 7 |
10 | eqid 2139 | . . . . . . . . . 10 | |
11 | 10 | negiso 8720 | . . . . . . . . 9 |
12 | 11 | simpri 112 | . . . . . . . 8 |
13 | 12 | imaeq1i 4878 | . . . . . . 7 |
14 | 9, 13 | eqtr3i 2162 | . . . . . 6 |
15 | 14 | supeq1i 6875 | . . . . 5 |
16 | 11 | simpli 110 | . . . . . . . . 9 |
17 | isocnv 5712 | . . . . . . . . 9 | |
18 | 16, 17 | ax-mp 5 | . . . . . . . 8 |
19 | isoeq1 5702 | . . . . . . . . 9 | |
20 | 12, 19 | ax-mp 5 | . . . . . . . 8 |
21 | 18, 20 | mpbi 144 | . . . . . . 7 |
22 | 21 | a1i 9 | . . . . . 6 |
23 | infrenegsupex.ss | . . . . . 6 | |
24 | 3 | cnvinfex 6905 | . . . . . 6 |
25 | 2 | cnvti 6906 | . . . . . 6 |
26 | 22, 23, 24, 25 | supisoti 6897 | . . . . 5 |
27 | 15, 26 | syl5eq 2184 | . . . 4 |
28 | df-inf 6872 | . . . . . . 7 inf | |
29 | 28 | eqcomi 2143 | . . . . . 6 inf |
30 | 29 | fveq2i 5424 | . . . . 5 inf |
31 | eqidd 2140 | . . . . . 6 | |
32 | negeq 7962 | . . . . . . 7 inf inf | |
33 | 32 | adantl 275 | . . . . . 6 inf inf |
34 | 5 | negcld 8067 | . . . . . 6 inf |
35 | 31, 33, 4, 34 | fvmptd 5502 | . . . . 5 inf inf |
36 | 30, 35 | syl5eq 2184 | . . . 4 inf |
37 | 27, 36 | eqtr2d 2173 | . . 3 inf |
38 | 37 | negeqd 7964 | . 2 inf |
39 | 6, 38 | eqtr3d 2174 | 1 inf |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2416 wrex 2417 crab 2420 wss 3071 class class class wbr 3929 cmpt 3989 ccnv 4538 cima 4542 cfv 5123 wiso 5124 csup 6869 infcinf 6870 cc 7625 cr 7626 clt 7807 cneg 7941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7718 ax-resscn 7719 ax-1cn 7720 ax-1re 7721 ax-icn 7722 ax-addcl 7723 ax-addrcl 7724 ax-mulcl 7725 ax-addcom 7727 ax-addass 7729 ax-distr 7731 ax-i2m1 7732 ax-0id 7735 ax-rnegex 7736 ax-cnre 7738 ax-pre-ltirr 7739 ax-pre-apti 7742 ax-pre-ltadd 7743 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-sup 6871 df-inf 6872 df-pnf 7809 df-mnf 7810 df-ltxr 7812 df-sub 7942 df-neg 7943 |
This theorem is referenced by: supminfex 9399 minmax 11008 infssuzcldc 11651 |
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